Forbidden Subgraphs Research Articles

The spectral Turán problem about graphs of given size with forbidden subgraphs

Let G be a graph of size m and ρ ( G ) be the spectral radius of its adjacency matrix. A graph is said to be H-free if it does not contain a subgraph isomorphic to H. Let C l 1 be the graph obtained by adding a pendent vertex to a cycle of length l. In this paper, we determine the graphs with the maximum spectral radius over C 3 1 -free graphs, and C 4 1 -free graphs with m ≥ 9 .

Forbidden Subgraphs Restricting Vertices of Degree Two in a Spanning Tree

Forbidden Subgraphs Restricting Vertices of Degree Two in a Spanning Tree

Line game-perfect graphs

The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $\chi_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, \[\chi_{[X,Y]}'(H)=\omega(L(H)),\] where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.

Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles

AbstractIn this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let denote the maximum number of edge‐disjoint copies of a fixed simple graph that can be placed on an ‐vertex ground set without forming a subgraph whose edges are from different ‐copies. The case when both and are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when and are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl, and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear ‐uniform hypergraph Turán problems to determine form a class of the multicolor Turán problem, following the identity , our results determine the linear hypergraph Turán numbers of every graph of girth 3 and for every up to a subpolynomial factor. Furthermore, when is a triangle, we settle the case and give bounds for the cases , as well.

A Note on Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.

SAT Modulo Symmetries for Graph Generation and Enumeration

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty-Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

Forbidden subgraphs in enhanced power graphs of finite groups

Forbidden subgraphs in enhanced power graphs of finite groups

On the girth of forbidden subgraphs of coloring graphs

The k-coloring graph of G, denoted Ck(G), is the graph whose vertex set is the proper k-colorings of the vertices of G, and where two k-colorings are adjacent if and only if they differ at exactly one vertex. A graph H is called permissible if there exists a graph G and a natural number k such that H is an induced subgraph of Ck(G), otherwise H is called forbidden. We show that every graph is either permissible or some subdivision of it is permissible. Furthermore, we prove if every edge in a triangle-free permissible graph is subdivided, the resulting graph is permissible, and all further subdivisions are also permissible. Although subdivisions tend to increase the girth of a graph, we conjecture that there exist forbidden subgraphs of arbitrarily large girth. Finally, we define forbidden subgraphs of girth seven and nine, the largest girths found so far.

On forbidden subgraphs of main supergraphs of groups

<p>In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.</p>

Paw-Type Characterization of Hourglass-Free Hamilton-Connected Graphs

This paper introduces the forbidden subgraph conditions for Hamilton-connected graphs. If the degree sequence of the graph is (4,2,2,2,2) and it is connected, then it is called hourglassΓ0. For integers i≥1, the graph Zi is paw, which is obtained by attaching one of the vertices of the triangle to one of the end vertices of a path with a number of edges i. We show that every graph G is Hamilton-connected if G is a Γ0-free, K1,3-free, Z14-free, and a 3-connected graph. Moreover, we give an example to show the sharpness of a paw-type forbidden subgraph in a 3-connected, Hamilton-connected graph. Our focus on the Hamilton-connected problem can be applied to data center networks (DCNs). In the future, we will remove the forbidden subgraph families from our conclusions when building the network to obtain the optimal communication cost. Our result extends the result of Ryjáček and Vrána (Discrete Mathematics 344: 112350, 2021).

On vertex Ramsey graphs with forbidden subgraphs

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs F, a graph A and a positive integer r, if every graph B∈F has a 2-vertex-connected subgraph which is not a subgraph of A, then there exists an F-free graph which is vertex r-Ramsey with respect to A. We prove that this sufficient condition for the existence of an F-free graph which is vertex r-Ramsey with respect to A is also necessary for large enough number of colours r.We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.

Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

A Complete Complexity Dichotomy of the Edge-Coloring Problem for All Sets of $$8$$-Edge Forbidden Subgraphs

A Complete Complexity Dichotomy of the Edge-Coloring Problem for All Sets of $$8$$-Edge Forbidden Subgraphs

Extremal results on feedback arc sets in digraphs

AbstractFor an oriented graph , let denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.

Some Characterizations and NP-Complete Problems for Power Cordial Graphs

A power cordial labeling of a graph G = V G , E G is a bijection f : V G ⟶ 1,2 , … , V G such that an edge e = u v is assigned the label 1 if f u = f v n or f v = f u n , for some n ∈ N ∪ 0 and the label 0 otherwise, and satisfy the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. The graph that admits power cordial labeling is called a power cordial graph. In this paper, we derive some characterizations of power cordial graphs as well as explore NP-complete problems for power cordial labeling. This work also rules out any possibility of forbidden subgraph characterization for power cordial labeling.

The Index of Signed Graphs with Forbidden Subgraphs

The Index of Signed Graphs with Forbidden Subgraphs

Existence of 2-factors in tough graphs without forbidden subgraphs

For a given graph R, a graph G is R-free if G does not contain R as an induced subgraph. It is known that every 2-tough graph with at least three vertices has a 2-factor. In graphs with restricted structures, it was shown that every 2K2-free 3/2-tough graph with at least three vertices has a 2-factor, and the toughness bound 3/2 is best possible. In viewing 2K2, the disjoint union of two edges, as a linear forest, in this paper, for any linear forest R on 5, 6, or 7 vertices, we find the sharp toughness bound t such that every t-tough R-free graph on at least three vertices has a 2-factor.

Decompositions of graphs of nonnegative characteristic with some forbidden subgraphs

A (d,h)-decomposition of a graph G is an order pair (D,H) such that H is a subgraph of G where H has the maximum degree at most h and D is an acyclic orientation of G−E(H) of maximum out-degree at most d. A graph G is (d,h)-decomposable if G has a (d,h)-decomposition. Let G be a graph embeddable in a surface of nonnegative characteristic. It is known that if G is (d,h)-decomposable, then G is h-defective d+1-choosable. In this paper, we investigate the (d,h)-decomposable graphs and prove the following four results. (1) If G has no chord 5-cycles or no chord 6-cycles, then G is (3,1)-decomposable, which generalizes a result of Chen, Zhu and Wang [Comput. Math. Appl. 56 (2008) 2073–2078]. (2) If G has no chord 7-cycles, then G is (4,1)-decomposable, which generalizes a result of Zhang [Comment. Math. Univ. Carolin, 54 (2013) 339–344]. (3) If G has no adjacent 3-cycles and no 6-cycles and no l-cycles for l∈{5,7}, then G is (2,1)-decomposable, which improves a result of Xu and Yu [Util. Math., 76 (2008) 183–189]. (4) If G has no 4-cycles and no 8-cycles, then G is (2,1)-decomposable, which improves a result of Li, Lu, Wang and Zhu [Discrete Applied Math., 331 (2023) 147–158]. We also show that some of these results are sharp.

Planar median graphs and cubesquare-graphs

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph G is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in G, and also in terms of subgraphs of G that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of G. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square-graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of “square-boundary” amalgamations. These considerations also lead to an O(nlogn)-time recognition algorithm to compute a decomposition of a planar median graph with n vertices into cubes and square-graphs.

Enumeration of Perfect Matchings of the Cartesian Products of Graphs

A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G \times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G \times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $G\times P_{2n+1}$ ($n\geqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G \times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G \times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G \times P_{n}$ by the eigenvalues of $G$.